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Chapter 4 Physical transformations of pure substance
For the pure substances, a phase diagram is a map of the pressure and
temperatures at which each phase is the most stable.
Chemical potential (µ): a property that is at the centre of discussion of
phase transitions (physical reactions) and chemical reactions.
4.1 The stability of phases:
A phase of a substance is a form of matter that is uniform throughout
in chemical composition and physical state. (solid, liquid, gas allotropes.)
A phase transition, the spontaneous conversion of one phase into
another phase, occurs at a characteristic temperature for a given pressure.
The transition temperature, Ttrs, is the temperature at which the two phase
are in equilibrium and Gibbs energy is minimized at the prevailing pressure.
A transition that is predicted from th
ermodynamics to be spontaneous may occ
ur too slowly to be significant in practice.
Exp. C(Diamond) C (graphite)
is too slow to be detected under ambient
condition.
Thermodynamically unstable phase that
persist because the transition is kinetically
hindered are called metastable phase.
4.2 Phase boundaries
The lines separating the regions, which are called phase boundaries,
shows the values of p and T at which two phases coexist in equilibrium.
Vapor pressure: the pressure of a vapor in equilibrium with the liquid.
Sublimation vapor pressure: the vapor pressure of the solid phase.
The vapor pressure of a substance increases with the increase of
temperature because at higher temperature the Boltzmann distribution
populates more heavily the sates of higher energy .
(a). Critical points and boiling points
When liquid is heated in an open vessel, at the temperature at which its
vapor pressure would be equal to the external pressure, vaporization can
occur throughout the bulk of the liquid. The condition of free vaporization
throughout the liquid is called boiling.
Boiling Temperature: the temperature at which the vapor pressure of a
liquid is equal to the external pressure.
At 1.0 atm: normal boiling point, Tb; At 1.0 bar, standard boiling point.
Boiling does not occur when a liquid in a closed vessel. Instead,
the vapor pressure, and the density of the vapor rise continuously as
increase of temperature.
The temperature at which the surface d
isappears is the critical temperature, Tc.
The vapor pressure at the critical temp
erature is called the critical pressure, pc.
At and above the Tc, a single uniform
phase called a supercritical fluid fills the c
ontainer and interface no longer exist.
At T > Tc, no liquid phase.
The Tc is the upper limit for liquid.
(b). Melting point and triple points
Melting point: the temperature at which, under a specified pressure, the liquid and
solid phases coexist in equilibrium. (Freezing point; liquid solid)
At p = 1 atm, normal freezing (melting) point, Tf
At p = 1 bar, standard freezing (melting) point.
Triple point (T3): a point three phase boundaries
meet. (three phases coexist in
equilibrium)
T3 occurs at a single definite p and T
characteristic of the substance.
thermodynamic temperature scale
The T3 marks the lowest pressure at which a
liquid phase of a substance can exist.
4.3 Three typical phase diagrams
(a)Water
The solid-liquid boundary has a
negative slope, which means a
decrease of volume on melting.
(dp/dT) < 0
At high pressures, different
structure forms of ice come into
stability as the hydrogen bonds
between H2O modified by
pressure.
Hydrogen-bonding structure in ice and water:
Ice (d = 0.9 cm3/g) Water (d = 1.0 cm3/g)
The volume decreases on melting. (ΔVm < 0)
(b) Carbon dioxide:
The positive slope of the solid-liqu
id boundary. (dp/dT >0)
The volume increases on melting.
The triple point lies above 1 atm. T
o obtain the liquid, it is necessary to
exert a pressure > 5.11 atm.
When a 67 atm CO2 squirts(噴出 )
to 1 atm, only the snow-like solid CO2
was obtained
(c). Helium
He behaves unusually at low temperatur
e.
The solid and gas phase are never in
equilibrium however low the temperature.
Solid helium can be obtained by applyin
g pressure (Helium is too light).
4He (I = 0), has a liquid-liquid phase tran
sition at its -line.
Helium-I phase behaves like a normal li
quid, Helium-II is a superfluid (no viscosit
y).
Phase Stability and Phase Transitions
4.4 The thermodynamic criterion of equilibrium
Chemical potential µ = Gm, µ is a measure of potential that a substance
for bring about physical or chemical change in a system.
Phase I
Phase II
The 2rd Law:
At equilibrium, the chemical potential of a
substance is the same throughout a sample,
regardless of how many phases are
present.
In equilibrium,
µ(phase I) = µ(phase II)
4.5 The dependence of stability on the conditions
dG = VdP – SdT ; dµ = VmdP – SmdT
(a). Temperature dependence of phase stability
(µ/T)p = – Sm
As the temperature raises, the che
mical potential of a pure substance decr
eases: Sm > 0 for all substance.
Sm(g) > Sm(l) > Sm(s)
The easiest way of bring about a pha
se transition is by changing the tempera
ture.
dG = dH – TdS – SdT = dU + pdV + Vdp – TdS – SdT
= dq – pdV + pdV + VdP – TdS – SdT = TdS + Vdp – TdS – SdT
= VdP – SdT
(b). The response of melting to applied pressure
For most substances except H2O, it is as thought the pressure is preven
ting the formation of the less dense liquid phase.
(µ/p)T = Vm
An increase in pressure increases the chemical potential of any substanc
e because Vm > 0.
In general, Vm(g) > Vm (l) > Vm (s)
The pressure dependence of the chemical potential (dG = VdP )
(a). Vm(l) > Vm(s) (b). Vm(l) < Vm(s)
p Tf p Tf
(c). The effect of applied pressure on vapor pressure
When pressure is applied to a condensed phase, its vapor pressure rises: in effec
t, molecules are squeezed out (壓出 ) the phase and escape as a gas.
Pressure can be exerted on the conden
se phases mechanically or by subjecting i
t to the applied gas of an inert gas (partial
pressure).
p = p*exp(VmΔp/RT)
Δp: the applied pressure
If VmΔp/RT << 1,
p p*(1 + VmΔP/RT)
(p – p*)/ p* = VmΔp/RT
Justification 4.1:
In equilibrium µ(l) = µ(g) dµ(l) = dµ(g)
dµ(l) = Vm(l)dP (applied pressure) ; dµ(g) = Vm(g)dp (p: vapor pressure)
The gas phase behaves as an ideal gas, dµ(g) = (RT/p)dp
RT/pdp = Vm(l) dP RT (1/p)dp (p* p) = Vm(l) dP (p* ΔP + p*)
If Vm(l) = constant, and vapor pressure change (p – p*) << ΔP
RT ln (p/p*) = Vm(l) ΔP
Illustration 4.1:
For water, the molar volume = 18.1 cm3mol-1, when subjected to an increas
e in pressure of 10 bar, the vapor pressure has an increase of 0.73 %.
4.6 The location of phase boundary
Phase boundaries – the pressure and temperature at which two phases are in
equilibrium. µ(p, T) = µ(p, T)
(a) The slope of phase boundaries
Phase boundaries dp/dT
On the phase boundaries, the two phase
continue to be in equilibrium.
dµ = dµ
For each pashe, dµ = – SmdT + Vmdp
– S,m dT + VmdP = – S,m dT + V,m dp
(V,m – V,m)dp = (S,m – S,m)dT
dp/dT = ΔtrsS/ΔtrsV
(b). The solid-liquid boundary:
dp/dT = ΔfusH/TΔfusV (ΔfusS = ΔfusH/T)
ΔfusV: the change in molar volume on melting
ΔfusV is usually positive and always small.
dp = (ΔfusH/ΔfusV) 1/T dT
p p* + ΔfusH/ΔfusV ln (T/T*)
When T is close to T*
(ln T/T* = ln(1+ (T – T*)/T*)
(T – T*)/T*
p p* + ΔfusH/T*ΔfusV (T – T*)
(c). The liquid-vapor boundary:
dp/dT = ΔvapH/TΔvapV (ΔvapH > 0 ; ΔvapV > 0 and large)
dp/dT > 0 for vaporization, and hence the boiling temperature is more respo
nsive to pressure than the freezing temperature.
Cooking in a covered pot,
Pressure boiling temperature cooking rate
Because the molar volume of a gas is larger than
that of a liquid.
dp/dT = ΔvapH/T(RT/p)
dlnp/dT = ΔvapH/RT2 (Clausius-Clapeyron eq.)
p = p*ex x = ΔvapH/R (1/T – 1/T*)
(d) The solid-vapor boundaries
dp/dT = ΔsubH/TΔsubV
dp/dT = ΔsubH/T(RT/p)
dlnp/dT = ΔsubH/RT2
p = p*ex x = ΔsubH/R (1/T – 1/T*)
Due to the ΔsubH > ΔvapH
The dp/dT slope for sublimation curve is st
eeper than that for vaporization curve at simi
lar temperature.
4.7 The Ehrenfest Classification of Phase Transitions:
Other transitions:
Solid-solid, conducting-superconducting and fluid-superfluid.
At the transition from a phase to another phase :
(µ/p)T – (µ/P)T = V,m – V,m = trsV
(µ/T)p – (µ/T)p = –S,m + S, m = trsS = trsH/Ttrs
The first derivatives of the chemical potentials with respect to pressure
and temperature are discontinuous at transition.
A transition for which the first derivative of the chemical potential with re
spect to temperature is discontinuous is classified as a first-order phase tr
ansition.
4.7 The Ehrenfest Classification of Phase Transitions:
1st
2rd
A second-order phase transition in the Ehrenfest sense is one in which the
first derivation of with respect to temperature is continuous but its second d
erivative is discontinuous.
V, H, and S do not change at the transition. The Cp is discontinuous at tran
sition but does not become infinite there.
A conducting-superconducting transition in metals at low temperature is an
example of second-order transition.
The term -transition is applied to a phase transition that is not first order y
et the heat capacity becomes infinites at the transition temperature.
This type of transition includes
order-disorder transitions in alloys,
the onset ferromagnetism, and the
fluid-superfluid transition of liquid
helium.
Second-order phase transition and -transitions:
Suppose the two shorter dimensions increases more than the longer
dimension when the temperature is raised.
The tetragonal cubic phase transition has occurs, but as it has not
involved a discontinuity in the interaction energy between atoms or the
volume they occupy, the transition is not first-order.
The order-disorder transition in -brass
(CuZn) is an example of a -transition.
At T =0 the order is perfect, but islands
of disorder appear as the temperature is r
aised.
The islands form because the transition
is cooperative in the sense that, once tw
o atoms have exchanged locations, it is e
asier for their neighbors to exchange their
location. The island grow in extent, and m
erge throughout the crystal at the transitio
n temperature (724 K).
5.1 Inorganic Solid-State Clathrate Compounds
5.1.1 Clathrate Hydrates:
5.1.1.1 Formation:
Many relatively nonpolar compounds that do not hydrogen bonding
with water are capable of forming clathrate hydrates.
In contrast, species such as alcohols, acids and ammonia, which are capable of
forming hydrogen bonds, do not generally form hydrates.
Solid clathrate hydrate are formed under very speci
fic temperature (often well above the melting point of
pure ice) and pressure. Indeed, some known gas hyd
rate are stable to 31.5 oC.
This high degree of thermal stability is currently a significant problem in the
natural gas industry. Transport of natural gas (CH4) in pipeline is plagued by
blockages caused by the formation of clathrate hydrates.
Solutions: 1. Drying and warming the gas.
2. Addition of large quantities of kinetic (PVP; polyvinylpyrrolidinone) and th
ermodynamic (methanol, glycol, salt solutions) hydrate formation inhibitors.
5.1.1.2 Structure:
Ice does not possess any cavities capable of including guest molecules.
Hydrates: a template reaction occurs in which polyhedral cavities are form acc
ording to guest size.
The cavities are composed
extensively of fused five- and six-
membered hydrogen-bonded rings.
512 (5: five-member ring, 12: the
number of ring in the cage)
Almost all clathrate hydrates form two basic structure termed type I and
type II.
Type I clathrate hydrates Type II clathrate hydrates
Guest molecule
In general, small guests occupy the sma
ll 512 type cavities. The small difference be
tween the size of this cavity in structures I
and II can have a significant effect on whic
h structural type is adopted.
He, H2 and Ne are small enough to diff
use through the cage and do not form hydr
ates.
In the real hydrates, some cages remain
empty. Thus the real hydrates have more
water than the “ideal composition”.
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